Description: Separation Scheme, which
is an axiom scheme of Zermelo's original
theory. Scheme Sep of [BellMachover] p. 463. As we show here, it
is
redundant if we assume Replacement in the form of ax-rep3444. Some
textbooks present Separation as a separate axiom scheme in order to show
that much of set theory can be derived without the stronger
Replacement. The Separation Scheme is a weak form of Frege's Axiom of
Comprehension, conditioning it (with ) so that it
asserts the
existence of a collection only if it is smaller than some other
collection that
already exists. This prevents Russell's paradox
ru2490. In some texts, this scheme is called
"Aussonderung" or the
Subset Axiom.

The variable
can appear free in the wff , which in
textbooks is often written . To specify this in the
Metamath language, we omit the distinct variable requirement ($d)
that
not appear in
.

For a version using a class variable, see zfauscl3456, which requires the
Axiom of Extensionality as well as Replacement for its derivation.

If we omit the requirement that not occur in , we can
derive a contradiction, as notzfaus3496 shows (contradicting zfauscl3456).
However, as axsep23455 shows, we can eliminate the restriction that
not occur in .

Note: the distinct variable restriction that not occur in
is actually redundant in this particular proof, but we keep it since its
purpose is to demonstrate the derivation of the exact ax-sep3454 from
ax-rep3444.

This theorem should not be referenced by any proof. Instead, use
ax-sep3454 below so that the uses of the Axiom of
Separation can be more
easily identified.